1. Binary, Hex, and Arbitrariness
CS/COE 0447
Jarrett Billingsley

2. Class Announcements
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3. Binary
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4. Positional number systems
the numbers we use are written positionally: the position of a digit within the number has a meaning
Most Significant
Least Significant
1,234
1 × 103 +
2 × 102 +
3 × 101 +
4 × 100 = 1s
10s
100s
1000s
100
101
102
103
how many digit symbols do we have in our number system?
10: 0, 1, 2, 3, 4, 5 ,6 ,7, 8, 9
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5. Range of numbers suppose we have a 4-digit numeric display
what is the smallest number it can show?
what is the biggest number it can show?
how many different numbers can it show?
= 10,000
what power of 10 is 10,000?
104
let's generalize. with n digits:
how many numbers can we represent?
10n
what is the largest number?
10n-1
- smallest is 0 (assuming no negatives…)
- biggest is 9,999
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6. 510 = 1012 Numeric Bases we use a base-10 (decimal) numbering system
but we can use (almost) any number as a base!
the most common bases when dealing with computers are base-2 (binary) and base-16 (hexadecimal)
when dealing with multiple bases, you can write the base as a subscript to be explicit about it:
510 = 1012
- you can have irrational bases! sure! why not!!!!!!
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7. Rules for Bases given base B, there are B digit symbols
each place is worth Bi, starting with i = 0 on the right
given n digits:
you can represent Bn numbers
the largest representable number is Bn – 1
so how about base-2?
how about with… 5 binary digits?
- base 2 has 2 digit symbols (0, 1)
- places are worth 2^0, 2^1, 2^2… = 1, 2, 4, 8, 16…
- with 5 bits:
- 2^5 (32) numbers
- max is 2^5-1 = 31
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8. Binary (base-2)
we call a Binary digIT a bit – a single 1 or 0
when we say an n-bit number, we mean one with n binary digits
1 × 128 +
0 × 64 +
0 × 32 +
1 × 16 +
0 × 8 +
1 × 4 +
1 × 2 +
0 × 1
= 15010
MSB LSB =
128s 64s 32s 16s s 4s 2s 1s
to convert binary to decimal: ignore 0s, add up place values wherever you see a 1.
- MSB and LSB are most/least significant bit
- since there are only 2 choices for each place, the process is super simple
- you gotta know these powers of 2! YOU GOTTA!
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9. Making change
you want to give someone $9.63 in change, using the fewest bills and coins possible. how do you count it out?
$5× $1× ¢× ¢× ¢× ¢×__
left:
$9.63 1 4 2 1 3
-$5=
$4.63
-$4=
$0.63
-50¢=
$0.13
-10¢=
$0.03
-0¢=
$0.03
-3¢=
$0.00
biggest to smallest
most significant to least significant
WHERE COULD THIS BE GOING...
- doooooes this process remind you of anything? ;O
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10. Converting decimal to binary
you want to convert the number 8310 to binary.
128s
64s
32s
16s 8s 4s 2s 1s
left: 83
- 0 = 83
- 64 = 19
- 0 = 19
- 16 = 3
- 0 = 3
- 0 = 3
- 2 = 1
- 1 = 1 1 1 1
for each place from MSB:
if place value < remainder:
digit = 1
remainder = remainder - place
else, digit = 0.
- see how binary makes this so simple: it reduces this "division" problem to a sequence of yes/no questions
- having only 0 and 1 makes a lot of things simpler… think about multiplication!
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11. Bits, bytes, nybbles, and words
8 bits (8b) = 1 byte (1B)
bit number: 7 6 5 4 3 2 1
1 bit
4 bits (4b) = 1 nybble
another nybble
a word is the "most comfortable size" of number for a CPU
BUT WATCH OUT:
Windows and x86 use word for 16 bits and double word (or dword) for 32 bits
when we say "32-bit CPU," we mean its word size is 32 bits
- we number bits from 0 at the LSB upwards to the left, since those are the powers of the place
- a byte is usually the smallest unit of measurement
- this is why your 50 megabit (Mb/s) internet connection can only give you at most 6.25 megabytes (MB) per second
- a 32-bit machine can, for example, add two 32-bit numbers at once
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12. Why binary? Whynary? cause it's the easiest thing to implement. :P
arithmetic also becomes really easy (as we'll see in several weeks)
so, everything on a computer is represented in binary.
everything.
- with binary, you only have to make circuits (or whatever) distinguish between two states
- technically everything is represented as "bit patterns" and we only treat it as "binary numbers" a lot of the time, but yanno
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13. Hexadecimal
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14. Shortcomings of binary and decimal
binary numbers can get really long, really quickly
3,927,66410 =
but nice "round" numbers in binary look arbitrary in decimal
= 32,76810
we could use base-4, base-8, base-16, base-32, etc.
base-4 is not much terser than binary
3,927,66410 =
base-32 would require 32 digit symbols
base-8 and base-16 look promising!
spoilers, no one uses base-8 anymore
- the "arbitrary look" of decimal is because 10 is not a power of 2 – there's that factor of 5 in there
- in C, C++, Java, I think C# and many others, a numeric literal that starts with 0 is in octal
- it was useful on 9-, 12-, 18-, and 36-bit machines
- but those haven't existed in decades
- DNA uses base 4 with a word size of 3 digits! tho we write the digits as "ACGT"
- each "codon" (word) of 3 digits represents one amino acid in a protein chain
- 64 codons map to ~22 amino acids, for some redundancy
- start and stop codons exist too – and likely all sorts of other things, like control flow (decision making)
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15. Let's make a base-2 16 number system
given base B,
there are B digit symbols
each place is worth Bi, starting with i = 0 on the right
given n digits:
you can represent Bn numbers
the largest representable number is Bn – 1
so how about base-16?
how about with 4 hex digits?
- there are 16 digit symbols (0-9, and then A-F)
- places are 16^0, 16^1, etc (1, 16, 256, 4096 – interestingly all end in 6 in decimal)
- 4 hex digits = 16^4 numbers = numbers
- biggest is 65535
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16. Hexadecimal or "hex" (base-16)
digit symbols A, B, C, D, E, F mean 10, 11, 12, 13, 14, 15
we call one hexadecimal digit a hex digit. no cute name :(
0 × 167 +
0 × 166 +
3 × 165 +
11 × 164 +
14 × 163 +
14 × 162 +
7 × 161 +
0 × 160 =
3,927,66410
003B EE70 =
to convert hex to decimal: use a dang calculator lol
- ACE are even, BDF are odd
- you can kiiiiinda do bytes in your head (2-digit hex numbers) since that only needs one multiplication by 16 but still
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17. 0xB8 = 1011 10002 = 18410 Getting a feel for it
4 bits (1 nybble) are equivalent to 1 hex digit
1 byte = 8 bits and therefore 2 hex digits (2 nybbles)
how many hex digits is a 32-bit number?
how many bits is a 5-digit hex number?
0xB8 = = 18410
(this is common notation for hex, derived from the C language.)
these are three ways of representing the same value.
they're different views of the same data.
- 32 bits = 8 hex digits
- 5 hex digits = 20 bits
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18. Converting between binary and hex
say we had this binary number:
starting from the LSB, divide into groups of 4 bits (pad with 0s if needed, like here)
Bin
Hex
0000
1000 8
0001 1
1001 9
0010 2
1010 A
0011 3
1011 B
0100 4
1100 C
0101 5
1101 D
0110 6
1110 E
0111 7
1111 F
know this table. 0x 3 B E E 7
hex → binary? look at the table!
- 4 bits = 1 hex digit = 4 bits, so all you need is this table to translate between em
- it's easy to recover the table if you forgot it: count in binary up to 15 and write the hex digits next to em
- I kinda remembered A=1010, and then counted up from there to remember lol
- since this is 6 hex digits, how many BYTES is it? (3)
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19. Powers of Two memorize at least the powers up to ~210 Dec Hex 20 1 0x122 4
0x4 23 8
0x8 24 16
0x10 25 32
0x20 26 64
0x40 27
128
0x80
Dec
Hex 28
256
0x100 29
512
0x200
210
1,024
0x400
211
2,048
0x800
212
4,096
0x1000
216
65,536
0x10000
220
1,048,576
0x100000
232
4.2 billion
uhhhhh
255 = 0xFF = 28-1 =
max 8-bit
1K (like kilobyte)
65535 = 0xFFFF = 216-1=
max 16-bit
1M (like megabyte)
0xFFFFFFFF = =
max 32-bit
- if you can't remember a power of two, add the previous one to itself!
- notice the hex versions are nice round numbers with a repeating pattern
- if you see numbers like this, you can visualize them as a 1 with 0s after
memorize at least the powers up to ~210
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20. Arbitrariness
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21. ok so everything's in binary, and hex is convenient shorthand
how does it know how to add two numbers?
how does it know how to manipulate strings?
how does it know if one pattern of bits is a string or a number or a video or a program or a file or an icon or
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22. IT DOESN'T
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23. What it means to be "arbitrary"
it means there's no reason for it to be that way.
we just kinda agree that it's how things are.
one of the biggest things I want you to know is:
what a pattern of bits means is arbitrary.
as a corollary:
the same pattern of bits can be interpreted many different ways.
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24. One bit pattern, many possible meanings
all information on computers is stored as patterns of bits, but…
how these bits are interpreted, transformed, and displayed is up to the programmer and the user
-59
Signed integer
R3G3B2 color
196
Unsigned integer
0xC4
Hexadecimal
call nz
z80 instruction Ä
Unicode
- you need CONTEXT to know what this value means
- even beyond the simple interpretation, its PURPOSE is another layer of abstraction on top of that
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25. The computer doesn't know or care.
when writing assembly (and C!) programs, the computer has no idea what you mean, cause nothing means anything to it
"my program assembles/compiles, why is it crashing?"
cause the computer is stupid
there's no difference between nonsense code and useful code
every "smart" thing a computer does, it does because a human programmed it to act like that
dumb CPU
its
code.
human.
lovely person"
- it's good at doing fun things with bit patterns
- but don't confuse what it does with intelligence
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